J Control Theory Appl 2011 9 (2) 273–277 DOI 10.1007/s11768-011-9015-8
Adaptive backstepping control and synchronization of a modified and chaotic Van der Pol-Duffing oscillator U. E. VINCENT 1,2 , R. K. ODUNAIKE 1 , J. A. LAOYE 1,3 , A. A. GBINDINNINUOLA 1 1.Department of Physics, Olabisi Onabanjo University, P.M.B. 2002, Ago-Iwoye, Nigeria; 2.Institute of Theoretical Physics,Technical University of Clausthal, Arnold-Sommerfeld-Str. 6, 38678 Clausthal-Zellerfeld, Germany; 3.Department of Physics, University of Ibadan, Ibadan, Nigeria
Abstract: In this paper, we propose a backstepping approach for the synchronization and control of modified Van-der Pol Duffing oscillator circuits. The method is such that one controller function that depends essentially on available circuit parameters that is sufficient to drive the two coupled circuits to a synchronized state as well achieve the global stabilization of the system to its regular dynamics. Numerical simulations are given to demonstrate the effectiveness of the technique. Keywords: Chaos; Synchronization; Back-stepping; Van der Pol-Duffing oscillator
1
Introduction
In 1990, the problem of chaos control [1] and synchronization [2] emerged as two exciting topics in nonlinear science that have promising applications. Pecora and Carroll [2] introduced a method to synchronize two identical chaotic systems evolving from different initial conditions, while Ott et al. [1] presented the OGY control algorithm. Chaos synchronization is closely related to the observer problem in control theory and recent studies deal with the synchronization problem based on control theory approach. Chaos synchronization has been intensively investigated in the context of many specific problems arising from physical [3, 4], chemical and ecological [5], and applications to secure communications [6∼8] to mention a few. Enormous research progress has been made in developing and understanding various types of synchronization schemes, such as adaptive control [9], active-backstepping design [10, 11], active control [12∼14], backstepping [15], and sliding mode control [16]. In most of the methods mentioned above, the controllers and the design approach are often very complex and could be difficult to achieve in practice. Thus, designing simple and available control inputs that can achieve global and stable synchronization of coupled oscillators is generally significant and of practical interest in view of the foreseen applications of chaos synchronization in circuits and lasers. This is an open challenge that has remained unresolved. In this paper, we propose an integrator and self-adaptive backstepping approach to deal with this problem using the modified van der Pol-Duffing oscillator circuit (MVDPD). In 2005 and 2007, Fotsin et al. [9, 17] proposed both nonadaptive and adaptive approaches to the synchronization problem of the MVDPD oscillator. The controllers proposed by Fotsin et al. [9, 17] contains feedback gains that are not dependent on the system parameters. In the control simulations, the feedback gains were arbitrarily selected to satisfy some stability criteria. This is definitely a limitation for experimental applications. In this paper, we show that 1) the synchronization control input can be very simple relative
to the systems being synchronized, and 2) the controller feedback gains could be strictly dependent on the system parameters and thus readily available for measurement. Furthermore, by considering the relationship between synchronization and control theory, we develop a nonlinear control scheme that is capable of driving the otherwise chaotic state to a regular one.
2 Modified Van der Pol-Duffing oscillator circuit The modified Van der Pol-Duffing oscillator circuit that we study here is described by the following set of normalized nonlinear ordinary differential equations [9, 17]: ⎧ 3 ⎨ x˙ = −m(x − αx − y + μ), (1) y˙ = x − y − z, ⎩ z˙ = βy − ηz. The circuit representation of equation (1) is shown in Fig. 1. When the load resistance RL placed in series with the inductor L shown in Fig. 1 is removed, the circuit reduces to the well-known Van der Pol-Duffing oscillator circuit described by the normalized equations: ⎧ 3 ⎨ x˙ = −m(x − αx − y), (2) y˙ = x − y − z, ⎩ z˙ = βy.
Fig. 1 Circuit diagrams of the modified Van der Pol-Duffing oscillator.
Received 29 January 2009; revised 7 November 2010. c South China University of Technology and Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2011
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The parameters m, α, and β are the circuit parameters and are chosen to ensure the chaotic behaviours of equations (1) and (2). μ is the offset term, and η is the parameter arising from the addition of the resistance RL in the modified circuit. The cubic term in equations (1) and (2) is derivable from nonlinear function of the nonlinear resistor N . x, y, and z correspond to the rescaled form of the voltage across C1 , the voltage across C2 , and the current through L, respectively. System (1) exhibit the one-scroll chaotic attractor shown in Fig. 2 (a) when α = 0.35, β = 300, μ = 0.035, η = 0.2, and m = 100, while system (2) exhibits a double-scroll chaotic attractor when α = 0.35, β = 300, and m = 300, as shown in Fig. 2 (b).
Fig. 2 Phase space of the Van der Pol-Dufing oscillators. (a) Double-scroll attractor of the Van der Pol-Dufing oscillators, (b) One-scroll attractor of the modified Van der Pol-Dufing oscillators with off-set term.
3
Synchronization control design
In the synchronization controller design, it is convenient to transform the modified Van der Pol-Duffing oscillator equation (1) by redefining the variables as follows: x = x3 , y = x2 and z = x1 ; expressing (1) in a drive-response configuration is given respectively as ⎧ ⎨ x˙ 1 = βx2 − ηx1 , (3) x˙ = x3 − x2 − x1 , ⎩ 2 x˙ 3 = −m(x33 − αx3 − x2 + μ), for the driver and ⎧ ⎨ x˙ 1 = βx2 − ηx1 , (4) x˙ = x3 − x2 − x1 , ⎩ 2 3 x˙ 3 = −m(x3 − αx3 − x2 + μ) − U (t), for the responding system, where U (t) is a nonlinear control input. We define the error system as the difference between the signals from drive and the response system as e1 = x1 − x1 , e2 = x2 − x2 ; and e3 = x3 − x3 . By considering the time derivative of the error signals together with equations (3) and (4), we obtain the error dynamics system: ⎧ ⎨ e˙ 1 = βe2 − ηe1 , (5) e˙ = e3 − e2 − e1 , ⎩ 2 3 − x ) + mαe + me + U (t). e˙ 3 = m(x3 3 2 3 3 With error dynamics (5), the synchronization problem is equivalent to that of realizing asymptotic stability of the zero solution to (5). In the absence of the controller U (t), equation (5) would have an equilibrium at (0, 0, 0), so that if appropriate U (t) is chosen such that the equilibrium (0, 0, 0) is unchanged, then asymptotic stabilization
would be realized, and hence, the synchronization between two systems (3) and (4) would be globally stable. To achieve this goal, we employ a self-adaptive procedure that involves three steps, wherein each step would require the stabilization of each subsystem in (5) using appropriately defined Lyapunov function. To start with, let V1 be a Lyapunov function for the e˙ 1 subsystem given as 1 (6) V1 = e21 , 2 and then, (7) V˙ 1 = e1 e˙ 1 = βe1 e2 − ηe21 . Suppose that e2 is a virtual control, then to stabilize e1 subsystem, let e2d = −e1 , where e2d implies a desired controller for e2 required to stabilize e1 subsystem. Thus, we obtain (8) V˙ 1 = −(β + η)e21 , which is negative definite. This implies that the subsystem, e˙ 1 in (5) is fully stabilized. To stabilize the second subsystem, in (5), let the error state between e2 and e2d be f2 , i.e., (9) f2 = e2 − e2d = e2 + e1 . Introducing another Lyapunov function V2 given as 1 (10) V2 = V1 + f22 , 2 and then, V˙ 2 = V˙ 1 + f2 f˙2 = V˙ 1 + f2 [e3 + (β − 1)e2 − e1 (1 + η)]. (11) Suppose that e3 is a virtual control in (11), and then, to stabilize the subsystem e2 and e1 , let (12) e3d = e1 (1 + η) − (β − 1)e2 . Thus, (13) V˙ 2 = V˙ 1 = −(β + η)e21 , which is negative definite. Let the error state between e3 and e3d be f3 , that is, (14) f3 = e3 − e3d . Introducing the third Lyapunov function V3 and its time derivative expressed as ⎧ 1 ⎨ V3 = V2 + f32 , (15) 2 ⎩ V˙ = V˙ + f f˙ . 3 2 3 3 3 V˙ 3 = V˙ 2 + f3 [m(e3 + αe3 + x3 3 − x3 ) + U (t)
− (βe2 −ηe1 )(1+η)+(β −1)(e3 −e2 −e1 )]. (16) If the controller U (t) is chosen such that (17) U (t) = k0 + k1 e1 + k2 e2 + k3 e3 , where the feedback gains are defined by ⎧ 3 3 ⎪ k0 = m(x3 − x3 ), ⎪ ⎨ k1 = (β − 1) − η(1 + η), (18) ⎪ k2 = (β − 1) + (1 + η)β − m, ⎪ ⎩ k3 = (1 − β) − mα, then, V˙ 3 = −(β + η)e21 (19) is negative definite, so that the two systems (3) and (4) are globally asymptotically synchronized. Note that the linear feedback gains (k1 , k2 , k 3 ) in controller (17) are essentially dependent on the circuit parameters and thus can be deter-
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mined directly without rigorous numerical simulations. On the other hand, the nonlinear component of U (t), namely, k0 depends on the nonlinear resistor (N ) in Fig. 1, which is also available, implying that our design control can be experimentally realized. Significantly, the nonlinearity of k0 imposes some advantages on the control input (17) because in practical physical systems, nonlinearities arise due to physical limitations, and their presence usually causes serious degradation of system performance and decrease in speed of response time. In some cases, they might cause chaotic perturbation to original regular behaviour if the controller is not well designed. This implies that the effect of nonlinearity cannot be completely ignored in the design and analysis of control inputs.
4
Stabilization control design
The goal of this section is to design a control input based on the backstepping scheme presented in the previous section that would stabilize the MVDPD oscillator unto its periodic orbit. To achieve this goal, we first reexpress the variables in equation (1) such that, x = z, y = y, z = x, and μ = 0. Thus, the controlled MVDPD oscillator is rewritten as ⎧ ⎨ x˙ = βy − ηx, (20) y˙ = z − y − x, ⎩ z˙ = −m(z 3 − αz − y) + u(t), where u(t) is a control function to be determined, and the offset term has been neglected. Considering the first equation in (20) and assume that y is a virtual control, we design a stabilizing function α1 (x) to make the time-derivative of the Lyapunov function x2 2 negative definite. Suppose that α1 (x) = −x, then, V1 (x) =
V˙ 1 (x) = −(η + β)x2 .
(21)
(22)
Let y¯ = y − α1 (x). Then, we obtain the (x, y¯) subsystem x˙ = β y¯ + (β − η)x, (23) y¯˙ = z + (β − 1)¯ y + [(β − 1) − (η + 1)] x. To stabilize the (x, y¯) subsystem (23), we design a stabilizing function α2 (x, y¯) for the virtual controlled variable z. By considering the following Lyapunov function for (23) defined as 1 (24) V2 (x, y¯) = V1 (x) + y¯2 , 2 and its corresponding time-derivative given by y + [(β − 1) − (η + 1)] x] V˙ 2 (x, y¯) = [z + (β − 1)¯ −(β + η)x2 , (25) and if we assume that α2 (x, y¯) = [(η + 1) − (β − 1)] x − (β − 1)¯ y, then we can make V˙ 2 (x, y¯) negative definite: V˙ 2 (x, y¯) = V˙ 1 (x) = −(β + η)x2 .
(26)
Suppose that the error state for the virtual control input α2 (x, y¯) is z¯ = z − α2 (x, y¯); then, we stabilize the full
(x, y¯, z¯) system given by ⎧ x˙ = β y¯ + (β − η)x, ⎪ ⎪ ⎨˙ y¯ = z + (β − 1)¯ y + βη x, 3 ˙ ⎪ − α¯ z + (βη α − 1)x z ¯ = −m{z ⎪ ⎩ +[α(β − 1) − 1]¯ y } + u(t),
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(27)
where βη = [(β − 1) − (η + 1)]. Note that in (27), we can leave the cubic product z 3 , which can also be expressed in terms of the variables x, y¯, z¯. Let the Lyapunov function for the full system (27) be given by 1 (28) V3 (x, y¯, z¯) = V2 (x, y¯) + z¯2 . 2 Then, its time derivative V˙ 3 (x, y¯, z¯) = V˙ 2 (x, y¯) + z¯z¯˙ is negative definite if z¯˙ = 0. This condition is fully satisfied if we choose the control input u(t) as follows: u(t) = k0 + k1 x + k2 y + k3 z, where
⎧ k0 ⎪ ⎪ ⎨ k1 ⎪ k2 ⎪ ⎩ k3
= mz 3 , = (β − 1) − η(1 + η), = (β − 1) + β(1 + η) − m, = (1 − β − mα).
(29)
(30)
Note that the control and the parameters in the controller (29) have the same form as that of equation (17). Whereas the feedback in (17) is provided by the error states, the feedback in (29) is provided by the state variables.
5 Numerical simulations In what follows, we now present numerical simulation results to verify the effectiveness of controllers (17) and (29). In all cases, we select the circuit parameters (m = 100, α = 0.35, β = 300, η = 0.2, μ = 0.035) such that the chaotic state in Fig. 2 (a) is maintained and choose the following initial conditions for the drive-response system x1 = 0.72, y1 = 0.04, z1 = −0.592, x2 = −0.85, y2 = −0.05, and z2 = −0.585. Moreover, from (18), using the above circuit parameters, we have (k1 , k2 , k3 ) = (298.76, 559, −334). In Fig. 3, we illustrate the performance of the controller given by equation (17) when activated at t 30. The error dynamics is found to convergence to the zero solution as t → ∞, implying that the synchronization between systems (3) and (4) has been achieved. Turning to the problem of stabilizing the dynamics to periodic solution, we activate controller (29) on the system (20). In Fig. 4, we illustrate the case in which the control is activated at t 30. Clearly, we see that the otherwise chaotic oscillation has been stabilized to the periodic orbit.
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tion of identical chaotic systems consisting of a drivenresponse system of modified Van der Pol-Duffing circuit oscillators. The method was also employed to drive the chaotic system to its regular dynamics. The proposed controller is essentially dependent on the measurable circuit parameters. Numerical simulations have been employed to verify the effectiveness of the proposed controller. The experimental realization of our proposed scheme remains to be presented, and we hope to report on this in the nearest future.
Acknowledgements U. E. Vincent is grateful to Olabisi Onabanjo University for granting Research Leave and Prof. Dr. Dieter Mayer for his hospitality during his stay as Humboldt Fellow. He acknowledges research support from the Alexander von Humboldt Foundation, Germany; the British Academy, the Royal Society of London and the Engineering and Physical Sciences Research Council, U.K., through the Newton International Fellowship. References Fig. 3 Error dynamics for the modified Van-der Pol Duffing oscillators when the control has been activated at t 0.
Fig. 4 Stabilization of the state space to periodic solutions. Controller (19) is activated at t 30.
6
Conclusions
In this paper, we have presented a simple self-adaptive synchronization technique that ensures global synchroniza-
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U. E. VINCENT et al. / J Control Theory Appl 2011 9 (2) 273–277 U. E. VINCENT was born in 1971. He is a lecturer in the Department of Physics, Olabisi Onabanjo University, Ago-Iwoye, Nigeria, and is currently with the Nonlinear Biomedical Physics Group at the Department of Physics, Lancaster University, U.K. He obtained his Ph.D. degree in Theoretical Physics, in 2005, from the University of Agriculture, Abeokuta, Nigeria. His main research interest is in the theory of dynamical systems with specific interest in bifurcations, control and synchronization. He applies these principles to several engineering and physical problems that arises in systems like the Josephson junctions, transport in nonequilibrium dynamical systems, among others. He has published his works in several reputable journals. In 2008, he won the Alexander von Humboldt Fellowship and in 2009, he also won the prestigious Newton International Fellowship-jointly awarded by the British Academy, the Royal Society of London; and the Engineering and Physical Sciences Research Council, U.K. Recently, he was awarded the 2nd Young African Mathematics Medal Award in Mathematical Physics (2009) by the African Mathematics Union (AMU). Corresponding author. E-mail:
[email protected]. R. K. ODUNAIKE was born in Gusau, Zamfara State of Nigeria. He received the B.S. degree in Engineering Physics (Materials Science option) from the University of Ife, Ile-Ife (now Obafemi Awolowo University) in 1981, and the Ph.D. degree in Solid state Physics from the University of Ibadan, Ibadan, Nigeria in 1993. He joined the services of the then Ogun State University (now Olabisi Onabanjo University (OOU)), Nigeria, in October 1985, as an assistant lecturer in the Department of Physics and rose to the rank of
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a Reader (associate professor) in 2005. He currently heads the Department and has similarly served the Department, Faculty and the University in various capacities. His research interests are in both theoretical and experimental solid state physics with particular reference to thin film/semiconductor devices characterization and environmental physics. He is also the Editor-in-Chief for the Journal of Physics and Environmental Science in the Department of Physics, OOU, Ago-Iwoye, Nigeria. E-mail:
[email protected]. J. A. LAOYE was born in 1964. He received his Ph.D. degree in 2001, from the University of Ibadan, Ibadan Nigeria. Since 1986, he has been a member of the Department of Physics, Olabisi Onabanjo University, Ago-Iwoye, Nigeria, where he is presently a senior lecturer and a member of the Statistical and Nonlinear Physics Group. His research interests are in the general area of Chaos synchronization and control, as well as nonlinear aspects of BoseEinstein condensates; and has published in reputable local and international journals. He is a member of IEEE and the Institute of Physics (IOP), Europe. E-mail:
[email protected]. A. A. GBINDINNINUOLA was born in 1983, and studied Physics with Electronics at Olabisi Onabanjo University, Ago-Iwoye, Nigeria, where she obtained her B.S. degree in 2008. Her interests are in the simulation and designs of electronic systems. E-mail:
[email protected]. —— ——————————————————————————————-
